Holant problems and counting CSP

Abstract

We propose and explore a novel alternative framework to study the complexity of counting problems, called Holant Problems. Compared to counting Constrained Satisfaction Problems (#CSP), it is a refinement with a more explicit role for the function constraints. Both graph homomorphism and #CSP can be viewed as special cases of Holant Problems. We prove complexity dichotomy theorems in this framework. Because the framework is more stringent, previous dichotomy theorems for #CSP problems no longer apply. Indeed, we discover surprising tractable subclasses of counting problems, which could not have been easily specified in the #CSP framework. The main technical tool we use and develop is holographic reductions. Another technical tool used in combination with holographic reductions is polynomial interpolations. The study of Holant Problems led us to discover and prove a complexity dichotomy theorem for the most general form of Boolean #CSP where every constraint function takes values in the complex number field C.

Algorithms and Computation

“…Another well studied framework is called H-coloring or Graph Homomorphism, which can be viewed as a special case of #CSP problems [4, 5, 14-16, 19, 20]. Recently, we proposed a new refined framework called Holant Problems [8,10] inspired by Valiant's Holographic Algorithms [25,26]. One reason such frameworks are interesting is because the language is expressive enough so that they can express many natural counting problems, while…”

“…One such class, called Holant * Problems [10], is the class of Holant Problems where all unary functions are freely available. If we allow only two special unary functions 0 and 1 as freely available, then we obtain the family of counting problems called Holant c Problems, which is even more appealing.…”

“…Previously a dichotomy theorem was proved for Holant * (F ), where F is any set of complex-valued symmetric functions [10]. It was used to prove a dichotomy theorem for #CSP in [10].…”

“…It was used to prove a dichotomy theorem for #CSP in [10]. For Holant c (F ) we were able to prove a dichotomy theorem valid only real-valued functions [10]. In this paper we manage to traverse in the other direction, going from #CSP to Holant Problems.…”

“…This is a new proof approach. Previously in [10], we reduced a #CSP problem to a Holant problem and obtained results for #CSP. Here, we go the opposite way, using results for #CSP to prove dichotomy theorems for Holant problems.…”